Since near the very beginning of electronic computing, Monte Carlo particle transport has been a fundamental approach for solving computational physics problems. Due to the high computational demands and inherently parallel nature of these applications, Monte Carlo transport applications are often performed in the supercomputing environment. That said, supercomputers are changing, as parallelism has dramatically increased with each supercomputer node, including regular inclusion of many-core devices. Monte Carlo transport, like all applications that run on supercomputers, will be forced to make significant changes to their designs in order to utilize these new architectures effectively. This dissertation presents solutions for central challenges that face Monte Carlo particle transport in this changing environment, specifically in the areas of threading models, tracking algorithms, tally data collection, and heterogenous load balancing. In addition, the dissertation culminates with a study that combines all of the presented techniques in a production application at scale on Lawrence Livermore National Laboratory's RZAnsel Supercomputer. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]